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December 1, 2013, 29(6): 788–797. http://www.neurosci.cn DOI: 10.1007/s12264-013-1385-0

·Method·

A novel single-trial event-related potential estimation method based on compressed sensing Zhihua Huang1,2,*, Minghong Li2,*, Shangchuan Yang2, Yuanye Ma2, Changle Zhou3 1

College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China

2

Kunming Institute of Zoology, CAS, Kunming 650223, China

3

Cognitive Science Department, Xiamen University, Xiamen 361005, China

*These authors contributed equally to this work. Corresponding author: Zhihua Huang. E-mail: [email protected] © Shanghai Institutes for Biological Sciences, CAS and Springer-Verlag Berlin Heidelberg 2013

ABSTRACT Cognitive functions are often studied using eventrelated potentials (ERPs) that are usually estimated by an averaging algorithm. Clearly, estimation of single-trial ERPs can provide researchers with many more details of cognitive activity than the averaging algorithm. A novel method to estimate single-trial ERPs is proposed in this paper. This method includes two key ideas. First, singular value decomposition was used to construct a matrix, which mapped singletrial electroencephalographic recordings (EEG) into a low-dimensional vector that contained little information from the spontaneous EEG. Second, we used the theory of compressed sensing to build a procedure to restore single-trial ERPs from this low-dimensional vector. ERPs are sparse or approximately sparse in the frequency domain. This fact allowed us to use the theory of compressed sensing. We verified this method in simulated and real data. Our method and dVCA (differentially variable component analysis), another method of single-trial ERPs estimation, were both used to estimate single-trial ERPs from the same simulated data. Results demonstrated that our method significantly outperforms dVCA under various conditions of signal-to-noise ratio. Moreover, the single-trial ERPs estimated from the real data by our method are statistically consistent with the theories of cognitive science. Keywords: compressed sensing; event-related

potentials; single-trial electroencephalography; singular value decomposition

INTRODUCTION Event-related potentials (ERPs) hidden in electroencephalographic recordings (EEG) are powerful indices of cognitive brain functions. When participants are exposed to specific external stimuli, potentials that are time-locked to the stimulus onset are evoked [1]. The EEG recorded under these conditions is commonly modeled as a linear combination of ERPs and spontaneous EEG. The most commonly-used method to estimate ERPs is to calculate an average across an ensemble of trials[2]; this is considered to eliminate spontaneous EEG activity. Obviously, the average cannot provide information on the trial-to-trial variability of ERPs. This variability is measurable [3] and meaningful in cognitive science [4] . For example, a study [5] of the P300 by single-trial analysis pointed out that people with schizophrenia have a smaller P300s; another study[6] based on single trials revealed different sub-types of response in late ERPs components; and EEG-fMRI single-trial coupling has been used to investigate the neuronal generators of the early posterior negativity in response to emotional auditory stimuli[7]. A previous study[8] also provided evidence that single-trial ERPs can be used to accurately classify the differences between experimental conditions. In their search for richer indicators of cognitive functions, researchers have been committed to finding better means by which single-trial ERPs can be estimated accurately.

Zhihua Huang, et al.

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A novel single-trial ERP estimation method based on compressed sensing

A number of techniques have been developed to estimate single-trial ERPs. The matched-filter proposed

optimization can then enable the recovery of such signals from very few measurements[32].

by Woody in 1967 is based on a signal model in which the

In this paper, we proposed a special single-trial ERPs

trial-to-trial change of amplitude and latency is depicted[9].

estimation method built upon the sparsity of ERPs in the

Subsequent researchers expanded Woody's thinking to

frequency domain. Elimination of spontaneous EEG is a

advance this field so that maximum likelihood could be used

difficult problem because it is always much stronger than

[10-14]

.

the ERPs. Almost all ERPs estimation methods assume

A method based on Bayesian theory[15] used second-order

that ERPs are irrelevant to spontaneous EEG. We also

a priori statistical information in the EEG to estimate single-

adopted this assumption in the process of constructing this

trial ERPs. Some researchers assume that EEG sources

new method. We started with this assumption to seek a

are independent, so Independent Component Analysis has

solution for eliminating spontaneous EEG by a de-noising

been used to estimate single-trial ERPs[16-18]. Another study

matrix obtained by singular value decomposition (SVD) of

proposed a classification-based framework and derived

spontaneous EEG. Further, a sensing matrix in CS was

a specialized algorithm for single-trial ERPs estimation

obtained by multiplying this de-noising matrix by the basis

to estimate the amplitudes and latencies for each trial

[19]

from this framework . On the other hand, analyses of the

of the inverse Fourier transform. Our method can be briefly

characteristics of signals in the frequency domain have

described in three steps: (1) obtain a de-noising matrix

been exploited to design procedures to estimate single-trial

by SVD of spontaneous EEG and a sensing matrix by

ERPs[20-26]. Wavelet and Fourier transformations usually

multiplying the de-noising matrix by that basis, (2) eliminate

underlie these procedures. Recently, a new approach

spontaneous EEG by the de-noising matrix obtained in the

based on the principle of projecting the signal and noise

first step, and (3) recover single-trial ERPs by the sensing

onto their respective signal and noise coefficient subspace

matrix using the theories of CS.

was reported[27]. In single-trial ERPs estimation and related fields,

METHODS

the sparsity of ERPs has drawn attention and been used to develop new methods of ERPs estimation and

Signal Model and Problem Formulation

representation. Sparse component decomposition has

A trial EEG recorded from the scalp is here written as

been applied to estimate ERPs . A framework based on

Ek

compressed sensing (CS), a special computing technology

number of channels, and T the number of sampling points

for sparse signals, was introduced for the representation of

for a trial. When EEG is recorded in the ERPs experimental

multichannel, multiple-trial EEG[28]. A classification method

paradigm, ERPs exist in the EEG, which is considered to

based on sparse representation of EEG signals and

be a linear combination of spontaneous EEG and ERPs.

ell-1 minimization has been suggested for building motor

So, we have

[24]

imagery-based brain-computer interface systems

[29, 30]

T×N

, where k represents the index of the trial, N the

Ek = Bk + Sk

. In

the present study, we started with the sparsity of ERPs in

(1)

where Bk denotes the spontaneous EEG of the kth trial

the frequency domain to build a novel method based on CS

and Sk the ERPs of the kth trial; certainly Bk, Sk

for single-trial ERP estimation.

We denote a column vector of E k , B k , S k as e, b, s,

CS was developed in a theoretical framework based on the theory of optimal recovery, the theory of n-widths, and

respectively. Clearly, e = b + s. The average, commonly (2)

attracted considerable attention in applied mathematics, that it may be possible to surpass the traditional limits

too.

used to estimate ERPs, can be expressed as

information-based complexity[31]. In recent years, CS has computer science, and electrical engineering by suggesting

T×N

where it is expected that

– =0 and S is the average

of sampling theory. CS builds upon the fundamental fact

of all Sk, the variability of which is ignored here. In actual

that we can represent many signals using only a few non-

settings, K usually ranges from 80 to 200. The broad

zero coefficients in a suitable base or dictionary. Nonlinear

applications of average estimation in cognitive science

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– show that S is not a bad approximation of Sk [1, 2, 33]. Seeking – to find every Sk, we consider that S is still valuable. k

k

Given the task is to estimate all S from E , what is known is clearly not enough. Most estimates of ERPs, including averaging and a variety of single-trial estimation methods[2, 9-26], build upon the assumption that ERPs are irrelevant to spontaneous EEG. Here, we also made this – assumption. Since S is an acceptable approximation of Sk – ~ and Bk is irrelevant to Sk, Bk= Ek – S can be considered an acceptable approximation of Bk. Furthermore, the column vectors of Bk are relevant to each other and they jointly reflect the status of spontaneous EEG in a trial. That is, we can find a projection from Bk, by which a column vector of Bk corresponding to a channel from the kth trial can be transformed to another vector in which most entries are very weak. This implies a solution to attenuate spontaneous EEG for estimation of single-trial ERPs. In the actual case, Bk is ~ unknown. It is reasonable to use Bk as a substitute for Bk. We explored the projection described above by SVD ~ ~ of Bk. According to the principle of SVD, we have Bk = U∆V, where U

T×T

, V

N×N

, and ∆ is a T × N rectangular

diagonal matrix with non-negative real numbers on the diagonal, which are called singular values. Each column vector of U corresponds to a singular value in ∆. When b is projected to a column vector of U, the singular value corresponding to it represents the power of b on it. Usually, the singular values decay rapidly if they are sorted in descending order. That is, the power of b concentrates on only a few dimensions if b is projected to the space determined by U. We can obtain a de-noising matrix Dk

M×T

,

where M
is obtainable, so y

M×1

method is built on CS. Sensing Matrix CS is a method for sparse signals with complete theories. T×1

Mathematically, a signal x

can be said to be m-

sparse when it has at most m non-zeros and m<
irrelevant to x, x

T×1

, and most entries of x are equal to

M×T , or near zero. To proceed, we defined Ak = DkΦ, Ak ~k k where D is derived from B being irrelevant to x, and Φ is

k

where e

problem of equation is still difficult. The next step of our

corresponds to a discrete inverse Fourier transform, Φ is (3)

k

T×1

Dk. Here, Dk is not a square matrix and not invertible. The

transformation, we may have s = Φx, where Φ

k

= Dks+Dkb

December 1, 2013, 29(6): 788–797

M×T

is a computable approximation

of D ks. Now, estimation of single-trial ERPs has been formalized as a problem of determining s by known y and

also irrelevant to x. Equation (3) can be further written as follows: y = Dke = Akx + η

(4)

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A novel single-trial ERP estimation method based on compressed sensing

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Algorithm 1. COSE, single-trial ERPs estimation lnput: EEG of K trials Ek, k = 1,…,K. Output: ERPs of channel j of K trials sk, k = 1,…,K. 1: initialize de-noising parameter M, sparsity level m, channel index j, inverse Fourier transformation matrix Φ. 2: 3: k = 1 4: while k< = K do – ~ ~ 5: e = COLj(Ek), Bk= Ek - S, Dk = SVM(B k), y = Dke, Ak = DkΦ. 6: x0 = 0 7: while stopping criterion is not met do 8: x1 = Hm(x0 + AT(y–Ax0)), x0 = x1. 9: end while 10: sk = Φx1 11: k = k + 1 12: end while

In equation (4), e is measured data, Dk is determined ~ via SV M(B k), y and A k are computable, and only x is to

zero except for the m entries of this vector with the largest

be determined. Clearly, the task in this study can be

either a limit on the number of iterations or a requirement

accomplished if x is determined well.

that y ≈ Ax1 in some sense.

In terms of other studies

[31, 32]

, x is an m-sparse signal

to be determined, Ak is a sensing matrix, η is noise, and y is the measurement of x. According to the theory of CS, for determining x it is required that M ≥ 2m. Always, m ranges from 10 to 20. EEG is usually recorded with 64, 128, or 256 electrodes, that is, N = 64, 128, or 256. M is slightly smaller than N. So, M ≥ 2m is completely met here. Signal recovery

magnitude. The stopping criterion in line 7 can consist of

dVCA Comparison between COSE and dVCA [11] will be made in the next section to demonstrate the advantages of our method. Here, dVCA is briefly described in this context. The dVCA algorithm consists of the following steps. (1) Choose the averaged result of EEG of K trials as

algorithms of CS are good for obtaining x from y and A .

the initial estimate of the ERPs component waveform.

Algorithm of Estimation

trial latency shift.

k

Generally, a variety of1 minimization or “greedy” algorithms for signal recovery in CS areas are applicable to our goal in the study. Here, a greedy algorithm named iterative hard thresholding (IHT)[32] was modified as the core of our singletrial ERPs estimation. This is called COSE, an abbreviation of compressed sensing, by which we emphasize the

(2) For all components and trials, estimate the single(3) Update the ERPs waveform component by component. (4) For all components and trials, estimate single-trial amplitudes. (5) If the stopping criterion is met, end. Otherwise, go to step (2).

importance of compressed sensing in our method. COSE is shown as Algorithm 1.

RESULTS

In Algorithm 1, COLj(·) represents an operator that takes the jth column of the matrix, Hm(·) denotes a hard

Simulated Data

thresholding operator on a vector that sets all entries to

We designed a procedure to build simulated EEG

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containing known ERPs by referring to the literature[11, 22, 24].

represents the standard deviation of simulated ERPs

In this procedure, simulated EEG was synthesized by the

and σn the standard deviation of the spontaneous EEG.

superposition of spontaneous EEG and simulated ERPs

We produced other 9 groups of simulated EEG for each

waveforms including several components. Eight university

participant by increasing the mean of amplitudes of the two

student volunteers (identified as S1 to S8) were randomly

components. They were –3 and 11, –5 and 13, –7 and 15,

selected in order to acquire spontaneous EEG. They were

–9 and 17, –11 and 19, –13 and 21, –15 and 23, –17 and

19–25 years old, half were female, and their hearing and

25, and –19 and 27. The SNRs of the other 9 groups were

sight were normal. A 64-electrode EEG instrument was

–8, –6, –4, –2, 0, 2, 4, 6, and 8.

used to record EEG at 1 000 Hz. The real 64-channel

We estimated single-trial ERPs from each simulated

EEG of participants in the resting state was viewed as

EEG by COSE and dVCA. Also, the average of each group

spontaneous EEG.

of simulated EEG was computed for comparison. Two

The spontaneous EEG of each participant was cut

randomly-selected examples of estimation results are given

into 72 segments, each of which spanned 400 ms. The

in Figure 1. Both examples showed that “averaged” is not

72 segments corresponded to 72 trials. A simulated ERPs

equal to “real”, which makes it clear that “averaged” is just

waveform containing two components is shown as the “real”

an approximation of single-trial ERPs. COSE resembled

curve on the right in Figure 1. A group of simulated ERPs

“real” much more than dVCA. This is an indication of the

waveforms for 72 trials were generated by adjusting the

advantage of COSE over dVCA.

amplitudes and latencies of the two components. In detail,

When the amplitudes are transformed to gray-scale,

the single-trial amplitudes of the two components were

a group of ERPs waveforms can be drawn as an image.

Gaussian-distributed with means –1 and 9 and standard

All estimation results of the two groups of simulated EEG

deviation 0.5, their latencies were Gaussian-distributed

are presented as gray-scale images in Figure 2. Visual

with means 168 and 274 and standard deviations 10 and

inspection showed that COSE images were much more

15. Superposition of spontaneous EEG segments and

similar to their real counterparts than dVCA images. These

simulated ERPs waveforms was carried out to construct

results demonstrated that COSE works much better than

a group of simulated EEG corresponding to 72 trials.

dVCA in the two groups of simulated EEG.

The signal-to-noise ratio (SNR) of this group of simulated

Next, we compared the real and estimated values of COSE and dVCA for the amplitudes and latencies of

EEG was –10 according to SNR = 20·log

, where σs

components of each simulated trial (Fig. 3). The data

Fig. 1. Examples of waveforms of real ERPs and ERPs estimated by averaging, COSE, and dVCA. The “real” curve is the actual ERPs waveform in simulated EEG. The “averaged” curve is the average of 72 trials. The “COSE” curve is the result estimated by COSE. The “dVCA” curve is the result estimated by dVCA. The left panel is from trial 52 of simulated EEG from S1 in which the SNR is –10. The right panel is from trial 69 of simulated EEG from S1 in which the SNR is –6.

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A novel single-trial ERP estimation method based on compressed sensing

793

Fig. 2. Groups of ERPs waveforms presented as gray-scale images by transforming amplitude to gray scale. Left panels, images from S1 with SNR = –10; right panels, S1 with SNR = –6. Upper panels, estimates by COSE; middle panels, the real waveforms; bottom panels, estimates by dVCA.

demonstrated that the effectiveness of estimating the

that SNR plays an important role in the performance of

latencies of both components by COSE was much better

single-trial ERPs estimation. Generally, the SNR of ERPs

than that of dVCA (Fig. 3, right panels; since both axes

in real EEG is low. The ability of a method to adapt to a low

are the same scale, good performance is indicated by

SNR is very important. The advantages of COSE over dVCA

clustering along the diagonal). For amplitude, we used

in the groups with a low SNR highlight important points.

different scales for the horizontal and vertical axes in order to contain as many dVCA results as possible (Fig. 3, left

Cognitive Experiment Data

panels). In this situation, better performance is indicated

Besides simulated data, our method was further verified

when data points cluster near the horizontal. The data

with real data from a cognitive experiment based on

showed that results of components estimated by COSE

sensory gating, which is a normal function of brain. The

were good, while dVCA was not satisfactory. These results

brain filters redundant information by this function to adapt

demonstrated the good performance of COSE and the

to a new environment and maintain the integrity of cognitive

advantage of COSE over dVCA.

function[35]. Sensory gating is related to cognitive processes

We built ten groups of simulated EEG for each of the

such as attention and working memory and works to protect

eight participants. All estimated results for amplitudes are

cognition[36]. The paired-click paradigm, a standard means

shown in Figure 4 and for latencies in Figure 5. The data

of researching sensory gating, was inserted into delayed-

showed that the correlations of real and estimated values

response tasks with different memory loads in three

of COSE were higher than those corresponding to dVCA

experiments. In Experiment 1, the participants were asked

in each group of all subjects for Components 1 and 2, and

to maintain a resting state. In Experiment 2, a randomly-

for amplitudes and latencies. Higher correlation implies

selected image of a face was set as the stimulus of a trial

better performance and the data showed the superiority of

and the participants were trained to maintain a state of

COSE over dVCA. In addition, a trend for the correlation to

low-load object working memory. In Experiment 3, two

increase with SNR was revealed for dVCA; this trend was

randomly-selected faces were set as the high memory load

also present for COSE though not obvious. This suggests

and the participants were also trained to maintain an object

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Fig. 3. Scatter diagrams of data from S1 with SNR = –10. Horizontal axis, real value; vertical axis, estimated value. Each point includes information of the real and estimated values of a component of a trial. Left panels, scatter diagrams of component amplitudes; right panels, scatter diagrams of component latencies. The title “COSE-Com1-Amp-Sca” denotes that this is a scatter diagram of the amplitudes of Component 1 estimated by COSE; other titles are similar.

working memory state, but with a high load. In Experiments

and the data sampling rate was 1 000 Hz; the reference

2 and 3, the participants were asked to remember the

electrode was on the nasion, and the resistances of all

faces and make a judgment as to whether the probe faces

electrodes were <5 kΩ. About 60 single-trial EEGs were

were the same as the target ones by clicking a computer

obtained in each session, since precoditioning usually

mouse. The three experiments corresponded to three

rejected some single trials.

different states of the participants. In each state, they were

COSE was applied to estimate single-trial ERPs from

all given stimuli of two consistent 10-ms sounds (frequency,

all real EEG. Subsequently, the amplitude of N100, a

1 000 Hz; intensity, 85 db; inter-stimulus interval, 500 ms;

component often measured in studies of sensory gating,

inter-test interval, 6-8 s; selected according to the paired-

was determined from each single-trial ERPs estimation and

click paradigm). EEG was recorded from the participants

matched according to the relation of the first sound and

in the different states. The real EEG was the superposition

second sound. The Wilcoxon signed-rank test was carried out on

of ERPs and spontaneous EEG. The three experiments

the pairs of amplitudes for all experiments on each participant.

consisted of 80 trials for each participant. Normally, the

As shown in Table 1, means corresponding to the 1st sound

ERPs evoked by the first sound is stronger than that by

were larger than those to the 2nd sound (P <0.05). This is

the second sound; i.e., sensory gating occurs. We verified

fully consistent with the principle of sensory gating[35]. The

COSE by judging whether the results in these real EEG

results of these cognitive experiments support the utility of

were consistent with this phenomenon.

COSE, our single-trial ERPs estimation method based on

Participants S1–S4 took part in Experiments 1

compressed sensing.

and 2, and S5–S8 in Experiments 1 and 3. During these experiments, NeuroScan was used to record and

DISCUSSIONS

precondition the EEG: the AC filter was set to 0.1–100 Hz,

ERPs have been important tools of research on cognitive

Zhihua Huang, et al.

A novel single-trial ERP estimation method based on compressed sensing

795

Fig. 4. Correlation of amplitudes. The correlation coefficient of real and estimated amplitudes was calculated for each component of each group of all subjects. The symbol ‘*’ represents the correlation coefficient of a group of Component 1 real amplitudes and amplitudes estimated by dVCA. The horizontal coordinate is the SNR of the group and the vertical coordinate the correlation coefficient. ‘°’, ‘+’ and ‘x’ have same implications, but correspond respectively to Component 1 estimated by COSE, Component 2 estimated by dVCA, and Component 2 estimated by COSE. Blue indicates Component 1 and red Component 2. The solid lines indicate COSE and the dotted lines dVCA. Each sub-graph displays the results from one participant (S1–S8).

Fig. 5. Correlation of latencies, as in Fig. 4.

functions for a long time. However, they are usually

difficult. Research on this issue can be traced back at least

estimated by averaging. Clearly, estimation of single-trial

to 1967[9], and a variety of methods have been proposed

ERPs can reveal much more about cognitive activities

and tried. In recent years, various ways to improve the

in the brain. However, estimation of single-trial ERPs is

estimation of single-trial ERPs have been developed[15, 19, 27].

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Table 1. Cognitive experiment results. Subject

Experiment 1

Experiment 2 or 3

1st sound

2nd sound

P value

1st sound

2nd sound

P value

S1

3.11 ± 2.1

1.93 ± 2.5

0.017

2.61 ± 2.5

1.19 ± 1.8

0.011

S2

2.95 ± 2.0

1.77 ± 2.3

0.011

2.53 ± 2.3

1.03 ± 1.9

0.010

S3

3.25 ± 2.3

1.65 ± 2.6

0.009

2.91 ± 2.6

1.02 ± 1.9

0.009

S4

3.11 ± 2.3

1.77 ± 2.7

0.012

2.92 ± 2.4

1.16 ± 1.7

0.013

S5

3.27 ± 2.2

1.72 ± 2.5

0.008

2.53 ± 2.6

0.76 ± 1.6

0.009

S6

3.15 ± 2.5

1.75 ± 2.4

0.009

2.37 ± 2.7

0.67 ± 1.7

0.011

S7

3.02 ± 2.1

1.73 ± 2.2

0.013

2.57 ± 2.5

0.62 ± 1.5

0.010

S8

3.12 ± 2.0

1.69 ± 2.6

0.015

2.39 ± 2.5

0.53 ± 1.8

0.009

The unit of measurement in the “1st sound” and “2nd sound” columns is μV. The data in “P value” columns are the P values of the Wilcoxon signedrank test on matched amplitudes of 1st and 2nd sounds.

Here, we sought a new solution to estimate single-

2011CB707800), National Natural Science Foundation of China

trial ERPs, starting with the fact that ERPs are sparse

(31271168), and Natural Science Foundation of Fujian Province,

or approximately sparse in the frequency domain.

China (2011J01344).

Compressed sensing, which recently emerged as a computing technology for sparse signals, was suitable for single-trial ERPs estimation in our research. We built a

Received date: 2012-11-16; Accepted date: 2013-01-31

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